We consider two algorithm for on-line prediction based on a linear model. The
algorithms are the well-known gradient descent (GD) algorithm and a new algorithm,
which we call EG?. They both maintain a weight vector using simple updates. For
the GD algorithm, the update is based on subtracting the gradient of the squared error
made on a prediction. The EG? algorithm uses the components of the gradient in the
exponents of factors that are used in updating the weight vector multiplicatively. We
present worst-case loss bounds for EG? and compare them to previously known bounds
for the GD algorithm. The bounds suggest that the losses of the algorithms are in
general incomparable, but EG? has a much smaller loss if only few components of the
input are relevant for the predictions. We have performed experiments, which show that
our worst-case upper bounds are quite tight already on simple artificial data.